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1. Say that p is prime, and p | a. Show that for some integer m,
p2 | ma.
2. Let X be a set with 5 elements, and Y a set with 0 elements.
Show that every function from X to Y is 1:1.
Yes, really. This question is written correctly, and means exactly what it says.
3. Let S be a subset of X, where S has m>0 elements and X has n elements.
We already showed that the number of functions from X to S is mn.
How many f:X->S have the property that f(s)=s for all s in S? Prove your answer.
(I'll leave the actual number in the comments, but then you still should think about proving it.)
4. Let X be a set. How big can X be, if
Each one of these has an answer, like "17 elements, no more". In which case give an example with 18 elements that doesn't have the property.
5. Let X={red,green,blue}, Y={0,1,2}, and Z={even,odd}. Let p:Y->Z be the parity function, so p(0)=p(2)=even, p(1)=odd. Describe the functions f:X->Y such that the composite p o f:X->Z is not onto. Show there are nine of them. How many of those f are 1:1? Draw three of them (put X, Y in columns, and draw arrows from each x to f(x)).
6. (The "Freshman's Dream".) Let a,b be integers. Show that
(a+b)3 is congruent to a3+b3 mod 3.
7. Is the same true with each 3 replaced by 4?
(I shouldn't have to add:) If true, prove it; if false, provide a counterexample.
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4 comments:
In #3, the number of functions is m ^ (n-m). But why?
Excuse me. But does "f(s)=s for all s in S" mean the "if x in X is not in S, there exists no mapping" or it means or a specific member s* in X should be mapped exactly to itself? I think they give different answers.
> Dear Professor Knutson,
> I tried to post a comment but I couldn't. :(
Weird!
> I have a question about #3. Does "f(s)=s for all s in S" mean the "if x in
> X is not in S, there exists no mapping" or it means or a specific member
> s* in X should be mapped exactly to itself? I think they give different
> answers.
It means neither. It means every s in S should be mapped to itself.
Your first interpretation seems to be that things not in S don't map anywhere. But for f to be a function, they have to map somewhere.
Your second interpretation would be correct if it said "there exists s in S such that f(s)=s". But it doesn't; it says "for all s in S, f(s)=s". Well, it was written in the opposite order, but this is how you should interpret it.
"i wanted to know if in problem # 1 the variable a is a natural number or an integer.
is it possible that a = p?"
Answers: an integer. And yes, the only condition was that a is a multiple of p. So a=p is certainly possible.
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